Coarse-graining non-stationary dynamics: linking early to late phases of zebrafish contests

Long-time behavior in complex systems, such as oceans, climate, and biological processes, is often coupled with non-stationary effects. In animal behavior, slow changes can occur without external stimuli (e.g. through the modulation of internal states such as satiety) and across multiple time scales, challenging their analysis from observed data. Here, we adopt a transfer operator approach to identify coarse-grained modes of non-stationary dynamics in the pair fighting behavior of zebrafish. Using high-resolution body point trajectories, we construct a discrete approximation of the transfer operator through a transition matrix over a waiting time τ. The dominant singular values of the transition matrix are related to the slowest relaxation times of a forward-backward process, and the corresponding left and right singular vectors detect initial and final coarse-grained behavioral states, respectively, "coherent pairs". By varying τ, we can extract coherent pairs across timescales, and we show that fight dynamics exhibit relaxation processes of at least the same duration as the fight itself. We find that the dominant coherent pairs are chasing maneuvers that evolve from those instigated by the eventual loser to those instigated by the winner.